Linear Spaces

Convergence

Definition: Let $(V,F)$ be a vector space. A sequence of vectors $\{x_n\}_{n=1}^{\infty}$ in $V$ is said to converge to a vector $x \in V$ if for every $\epsilon > 0$ there exists an integer $N$ such that $\|x_n - x\| < \epsilon$ for all $n \geq N$. In this case we write $x_n \rightarrow x$ as $n \rightarrow \infty$.

Remark: The sequence is said to be convergent if it converges to some vector $x \in V$. Otherwise, it is said to be divergent.

Example: Let $V=R \ \ \|v\|=v$, consider the sequence

$$\left\{ (\frac{1}{2})^n \right\}_{n=1}^{\infty}$$

Proove that this sequence converges to $0$ as $n \rightarrow \infty$.

$~$Proof: DO IT LATER

Definition: Let $(V,F, \|.\|)$ be a normed space. A sequence of vectors $\{x_n\}_{n=1}^{\infty}$ in $V$ is said to be a Cauchy sequence if for every $\epsilon > 0$ there exists an integer $N$ such that $\|x_n - x_m\| < \epsilon$ for all $n,m \geq N$.

• Proof: $\|x_n - x_m\| < \epsilon$ for all $n,m \geq N$ implies $\|x_n - x\| < \epsilon$ for all $n \geq N$.

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#EE501 - Linear Systems Theory at METU